We construct free abelian subgroups of the group U(AΓ) of untwisted outer automorphisms of a right-angled Artin group, thus giving lower bounds on the virtual cohomological dimension. The group U(AΓ) was studied in [5] by constructing a contractible cube complex on which it acts properly and cocompactly, giving an upper bound for the virtual cohomological dimension. The ranks of our free abelian subgroups are equal to the dimensions of principal cubes in this complex. These are often of maximal dimension, so that the upper and lower bounds agree. In many cases when the principal cubes are not of maximal dimension we show there is an invariant contractible subcomplex of strictly lower dimension.

中文翻译：

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RAAGs的自同构群的立方复形和阿贝尔子群

我们构造群的自由阿贝尔子群ü(一种Γ) 的直角 Artin 群的未扭曲外自同构，从而给出了虚拟上同调维数的下界。群组ü(一种Γ) 在 [5] 中通过构造一个可收缩的立方体复合体进行了研究，在该复合体上它正确地和协紧地作用，给出了虚拟上同调维度的上限。我们的自由阿贝尔子群的秩等于主立方体在这个复杂的。这些通常是最大维度的，因此上限和下限一致。在许多情况下，当主立方体不是最大维数时，我们表明存在严格低维的不变可收缩子复合体。